= Solution
For each interaction neighbourhood, set
$$
G_s^{(Z)}=\int_{\mathbb R}w(t)
e^{itH_s}h_Ze^{-itH_s}\,dt.
$$
Then the shell term is $g^{(Z,s)}=G_s^{(Z)}-G_{s-1}^{(Z)}$. Since $H_0=h_Z$, its conjugation leaves $h_Z$ fixed, so $G_0^{(Z)}=h_Z\int w=h_Z$. Since $H_D=H$, $G_D^{(Z)}$ is precisely the choice of $g^{(Z)}$ from (b). Therefore the <telescoping local-shell decomposition> gives
$$
\boxed{h_Z+\sum_{s=1}^Dg^{(Z,s)}
=G_0^{(Z)}+\sum_{s=1}^D
(G_s^{(Z)}-G_{s-1}^{(Z)})
=G_D^{(Z)}=g^{(Z)}.}
$$
No limiting interchange is needed: the sum has finitely many shells. Each $G_s^{(Z)}$ is supported within the union of supports appearing in $H_s$ together with $Z$, because its <unitary time evolution> acts trivially outside that neighbourhood. The individual shell terms need not commute with the full <ground state> projector; the full filtered sum does.
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